Decide whether each of the following is true or false without using a calculator:
The problem is:

$$11^{99}\equiv 1\pmod{5}$$

Now I know I can break the $11$ into $(10+1)^{99}$ and maybe rewrite it as $(10+1)^{98}\times (10+1)$
and then realize that $10+1\equiv 1\pmod{5}$ and then just deal with $(10+1)^{98}$ but that means I have to do this $97$ times until at the end I have $10+1\equiv 1\pmod 5$ but I feel like this is a very stupid way to look at it. Is there a little trick of sorts that I can use on problems like this when the exponents are huge?

$\begingroup$Not necessary. $11$ raised to the power of itself any number of times ends with $1$, take away, $1$ from it, it ends with $0$, so it's divisible by $5$$\endgroup$
– GohP.iHanApr 21 '15 at 4:32

$\begingroup$@GohP.iHan, the OP further asked if there's any method in general that'll allow him to approach problems like this. That's precisely the reason I mentioned these theorems in my answer.$\endgroup$
– Prasun BiswasApr 21 '15 at 5:33